This literature review provides a brief discussion of the Riemann Hypothesis, a conjecture regarding the location of the zeros on the Riemann zeta function.  We also look at the implications of solving the hypothesis, and investigate historical attempts to solve the conjecture

Allen, D.

Bonetta-Martin, K.

Codling, E.

Jefferies, S.

Plymouth Electronic Archive and Research Library

The Plymouth Student Scientist, 2016, 9, (2), 249-257The Riemann HypothesisDanny Allen, Kyle Bonetta-Martin, Elizabeth Codling and SimonJefferiesProject Advisor: Tom Heinzl, School of Computing, Electronics and Mathematics,Plymouth University, Drake Circus, Plymouth, PL4 8AAAbstractThis literature review provides a brief discussion of the Riemann Hypothesis, a con-jecture regarding the location of the zeros on the Riemann zeta function. We alsolook at the implications of solving the hypothesis, and investigate historical attemptsto solve the conjecture.[249]The Plymouth Student Scientist, 2016, 9, (2), 249-2571 Introduction1.1 The Riemann HypothesisThe Riemann Hypothesis is a conjecture regarding the zeros of the Riemann zetafunction. First proposed by Bernhard Riemann in 1859 [1], the hypothesis has yet tobe proved despite great efforts for over 100 years. The hypothesis remains one ofHilbert’s unsolved problems [2], as well as being a Millennium Prize Problem [3].To fully understand the Riemann Hypothesis we must first introduce the Riemannzeta function, ζ(s). The Riemann zeta function is defined by the seriesζ(s) =∞∑n=11ns=11s+12s+13s+ . . . , (1)for s ∈ C with <(s) > 1. An equivalent form of the zeta function is given by the Eulerproduct formula, valid for <(s) > 1:ζ(s) =∏p prime11− p−s=11− 2−s· 11− 3−s. . .11− p−s. . . . (2)As a convergent infinite product cannot converge to zero (because its logarithm is aconvergent series), this form of the zeta function shows that the zeta function doesnot vanish for <(s) > 1.Another equivalent definition for the zeta function in integral form is given by:ζ(s) =1Γ(s)∫ ∞0x s−1ex − 1dx . (3)In order to use zeta function to explain the Riemann Hypothesis, we must first extendthe domain of the function to all complex values of s – through the method of ana-lytic continuation. First, we define the Dirichlet eta function, which converges for anycomplex number with <(s) > 0, and is given by the following Dirichlet series:η(s) =∞∑n=1(−1)n−1ns=11s− 12s+13s− 14s+ . . . . (4)We can now extend the Riemann zeta function to the domain where <(s) > 0through the following relation between the eta and zeta functions. The eta functioncan be represented by(1− 22s)ζ(s) =∞∑n=11ns−∞∑n=12(2n)s=11s+12s+13s+14s+ . . .− 22s− 24s− 26s− 28s− . . .=11s− 12s+13s− 14s+ . . . =∞∑n=1(−1)n−1ns= η(s).(5)[250]The Plymouth Student Scientist, 2016, 9, (2), 249-257The series on the right-hand side converges when <(s) > 0, so we have nowextended the domain of the zeta function to values of s with positive real parts asidefrom the zeros of 1 − 2/2s at s = (1 + 2πin)/ ln 2. The zeta function can be extendedto these values as well by taking limits leaving only a simple pole at s = 1 [4].To extend the domain as Riemann did himself in his 1859 paper [1], we nowestablish the following functional equation - and therefore analytically continue thedomain of the zeta function to all values of s excluding a simple pole at s = 1 withresidue 1. The functional equation, as explained by Titchmarsh [4], is:ζ(s) = 2sπs−1 sin(πs2)Γ(1− s)ζ(1− s). (6)From the functional equation, we can now see that the zeta function will vanish fornegative even integers due to the cyclical nature of the ‘sin(πs/2)’ component of thefunctional equation. These values are known as the trivial zeros of the zeta function.The situation is different for the positive even integers, for which Γ(1− s) has a simplepole such that sin(πs/2)Γ(1− s) is regular and non-zero for positive even s.Hadamard in [5] and de la Vallée-Poussin in [6] both proved that there are nozeros on the line <(s) = 1 . From this, we see that all non-trivial zeros must lie inthe critical strip where s has real part between 0 and 1. The Riemann Hypothesisis the conjecture which states that the non-trivial zeros of the zeta function all lie onthe critical line, where <(s) = 1/2. As the Riemann zeta function is a map from CFigure 1: Maple plot of the Riemann zeta function where <(s) = 12. Solid and dashedlines represent the real part and imaginary parts of ζ(s), respectively.to C, hence returns complex values upon complex input, a plot of the entire functionwould require a four-dimensional space. Thus, in Figure 1, we separately show thereal and imaginary values of the zeta function for <(s) = 12. When both lines intersectthe horizontal axis, say at t0, the zeta function is zero at s = 1/2± t0. We can see thatthe first three zeros have imaginary parts approximately given by ±14, ±21, and ±24.The zeros are symmetric about the real axis asζ(s∗) =∞∑n=11ns∗=∞∑n=11(ns)∗=∞∑n=1(1ns)∗= ζ(s)∗ , (7)where a∗ denotes the complex conjugate of a.[251]The Plymouth Student Scientist, 2016, 9, (2), 249-2571.2 Generalised Riemann Hypothesis and Extended RiemannHypothesisThe Riemann zeta function is of the formLk(s,χ) ≡∞∑n=1χk(n)ns, (8)where χk(n) is a function defined on Z and has a period k . We see that if χk(n) = 1for all n, then this is equivalent to the Riemann zeta function. If a function has thisform, then it is a Dirichlet L-series, see [7]. While the Riemann Hypothesis onlyrelates to the zeroes of the Riemann zeta function, it can be generalised to all DiricletL-series. The Generalised Riemann Hypothesis (GRH) states that the real parts of allzeroes of any Dirichlet L-series are not larger than 1/2 [7].The Extended Riemann Hypothesis (ERH) is the Riemann Hypothesis formulatedfor Dedekind zeta functions. The Dedekind zeta function is defined as follows:ζK (s) =∑a1(Na)s, (9)where K is a number field where OK is the integral closure of Z in K , a is a non-zero ideal of OK , and Na is the norm of a. The ERH states that when ζk = 0 and0 < <(s) < 1, then <(s) = 1/2 [8].It is important to note that either of these implies the Riemann Hypothesis as aconsequence.2 Implications of the Riemann HypothesisThere are many theorems in mathematics that assume the Riemann Hypothesis:thus, if someone were to prove the hypothesis, it would have large implications through-out mathematics. For example, the Miller-Rabin primality test – an algorithm that de-termines if a given number is prime – relies on the Extended Riemann Hypothesisbeing true[9]. If the ERH is not true then this, and many other theorems similar, be-come null and void. There are also theorems that assume the hypothesis is not true,such as the Gauss class number problem, which relies on the Generalised RiemannHypothesis being false [10].Perhaps more importantly, the Riemann Hypothesis is vital in improving the errorterm in the prime number theorem [11]. This theorem attempts to estimate the rate atwhich prime numbers appear, or more specifically, the rate at which they become lesscommon. The prime number theorem states that the prime-counting function, π(x),can be approximated using x/ ln(x) and was first proposed by Gauss and Legendretowards the end of the 18th century. The function π(x) is defined as the number ofprime numbers below x , for example π(20) = 8, as there are 8 primes below 20 [12].[252]The Plymouth Student Scientist, 2016, 9, (2), 249-257A slightly better approximation of π(x) can be given by π(x) ≈ Li(x), whereLi(x) =∫ x21ln tdt (10)denotes the (offset) logarithmic integral function. This improvement was demon-strated independently by Hadamard and de la Vallée-Poussin in 1896 using the Rie-mann zeta function as a main part of the proof, specifically the fact it is non-zero forall complex values of s that have the form s = 1 + i t with t > 0. Three years later, dela Vallée-Poussin in [13] proved thatπ(x) = Li(x) +O(xe−a√ln x), (11)and this has since been improved in [14] to:π(x) = Li(x) +O(xe−A(ln x)0.6/(ln ln x)0.2). (12)Assuming the Riemann Hypothesis the error term in this equation can be simpli-fied and reduced toπ(x) = Li(x) +O(√x ln x), (13)proved by von Koch in 1901 [15]. Littlewood also used the Riemann Hypothesis toprove that Li(x) − π(x) switches from positive to negative infinitely many times forsufficiently large x [11].Essentially, the Riemann Hypothesis implies a reduced error term for π(x) so thatwe can better understand the distribution of prime numbers.3 Attempts at Proving the Riemann Hypothesis3.1 Equivalent StatementsMany have tried to prove the Riemann Hypothesis, but (thus far) without success. Oneway of looking at the problem is to find statements equivalent to the hypothesis, orstatements that would imply the hypothesis. Here, we will show a couple of examplesof equivalent statements to the Riemann Hypothesis – all providing a different way toapproach the proof (or, indeed, disproof) of the hypothesis.The asymptotics of the prime counting function given in (13), for instance, is equiv-alent to the Riemann Hypothesis, but there are many other equivalent statements.Another example is the following [16]: firstly, define the sum of divisors of n asσ(n) =∑d/nd ; (14)for example, σ(12) = 1 + 2 + 3 + 4 + 6 = 16. Introducing Euler’s constant,γ = limx→∞n∑k=11k− ln n ≈ 0.577215664 , (15)[253]The Plymouth Student Scientist, 2016, 9, (2), 249-257one can state that the Riemann Hypothesis is equivalent to the inequalityσ(n) < eγn ln ln n , (16)when n > 5040.This is Robin’s Theorem [17]. The inequality holds for most integers n but is falsefor 2, 3, 4, 5, 6, 7, 8, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360,720, 840, 2520 and 5040 [18].Robin’s Theorem states that if this inequality holds for all n > 5040, then theRiemann Hypothesis is true.Lagarias continued from Robin’s work in [19], and created an inequality equivalentto the Riemann Hypothesis with a smaller bound on n, using the nth harmonic number,Hn =∑nj=1 1/j , to state thatσ(n) ≤ eHn lnHn + Hn (17)is equivalent to the Riemann Hypothesis with the weaker condition n ≥ 1. Neverthe-less, Robin’s Theorem is often preferred for analysis as it does not contain harmonicnumbers.These and other equivalent statements help us to understand the many facetsand implications of the Riemann Hypothesis.3.2 Other Attempts at ProofOf course, the finding, and proving, of an equivalent statement is not the only attemptused to determine the validity of the Riemann Hypothesis. As for its disproof, themost obvious tactic is disproof by counterexample: calculate the zeros of the function,and if a zero appears that is neither trivial nor on the critical line, then the RiemannHypothesis cannot be true. Many – over 1013 – zeros of the zeta function have beencalculated [20], and as yet all have been trivial, or on the critical line of <(s) = 1/2.There has been some progress made on proving the hypothesis by ‘narrowingdown’. For example, it has been proved that there are infinitely many zeros of thezeta function along the line <(s) = 1/2 – the critical line – by Hardy, in 1914 [21].Later, in 1989, Conrey proved that at least 40% of all zeros lie on the critical line [22].This was continued in 2011, when Bui et al. increased this to 41.05% [23].Others have seen relations to the Riemann Hypothesis from other, often seem-ingly unrelated, topics. For example, the Lee-Yang theorem of statistical mechanicsimplies that (for certain statistical field theories) all the zeros associated with the par-tition function of the model lie on a critical line [24]. Although there has not been anyprogress on solving the Riemann Hypothesis, this relation – of all (bar trivial) zeroslying on a particular line – may be one of the tools for finding a proof.[254]The Plymouth Student Scientist, 2016, 9, (2), 249-2573.3 The Hilbert-Pólya ConjectureThe Hilbert-Pólya Conjecture is an unpublished idea accredited to David Hilbert andGeorge Pólya about a possible way to solve the Riemann hypothesis. Pólya, in aconversation with Landau in 1914, remarked that the Riemann hypothesis would betrue “if the non-trivial zeros of [the Riemann zeta function] were so connected with thephysical problem that the Riemann hypothesis would be equivalent to the fact that allthe eigenvalues of the physical problem are real.” [25].This idea translates to finding a self-adjoint operator A such that the eigenvaluesof A are the same as the imaginary parts of the zeros of the zeta function. As anyself-adjoint operator has only real eigenvalues, then if ζ(s) could be mapped to A,then there could not be a zero off of the critical line, and so the Riemann Hypothesiswould be proven.This conjecture was interesting when it was proposed, but there was little evi-dence in 1914 to support the idea that there would be such a corresponding operator.However, Selberg in [26] showed a connection between the length spectrum of a Rie-mann surface and the eigenvalues of its Laplacian, typically a self-adjoint operator.The length spectrum of the Riemann surface is the set of closed geodesics (straightlines on the surface) ordered by the length of the line. This connection, the Selbergtrace formula, is very similar to the explicit formula connecting the Riemann zeta func-tion and prime powers. This discovery gave credence to the Hilbert-Pólya conjecture,although what the self-adjoint operator could be is still unknown.The first written instance of the Hilbert-Pólya conjecture appeared in [27], whereMontgomery was looking at the spacing between the zeros. In this paper, he notesthat the pair correlation function he derives for the spacing between zeros has thesame statistical distribution as the eigenvalues of a random complex Hermitian matrix[28]. A Hermitian matrix is one that is self-adjoint, and so this observation also lendscredence to the Hilbert-Pólya conjecture.4 ConclusionThe Riemann Hypothesis is such an important unsolved problem – not only due to itsfamed difficulty, but also due to the huge impact its proof or disproof would make onmathematics. Many proofs have been attempted from varying fields and disciplines,including many amateur mathematicians trying their hand for the million dollar millen-nium prize. As of now, there is no definitive proof either way, though with the sheeramount of zeros calculated – none disagreeing with Riemann’s hypothesis, and theamount of mathematics that rely on the hypothesis, it seems plausible, at the momentat least, to be true.[255]The Plymouth Student Scientist, 2016, 9, (2), 249-257References[1] Bernhard Riemann. Ueber die Anzahl der Primzahlen unter einer gegebenenGrösse. Ges. Math. Werke und Wissenschaftlicher Nachlaß 2, 145–155, 1859.[2] Wikipedia, Hilbert’s problems, accessed 16/12/2015,https://en.wikipedia.org/wiki/Hilbert’s problems.[3] Enrico Bombieri. Problems of the millennium: The Riemann hypothesis, 2000.[4] Edward Charles Titchmarsh and David Rodney Heath-Brown. The theory of theRiemann zeta-function. Oxford University Press, 1986.[5] Jacques Hadamard. Sur la distribution des zéros de la fonction ζ(s) et sesconséquences arithmétiques. Bulletin de la Societé mathematique de France24, 199–220, 1896.[6] Charles-Jean De la Vallée Poussin. Recherches analytiques sur la théorie desnombres premiers. Annales de la Societe Scientifique de Bruxelles 21 B, 351–368, 1896.[7] Wolfram MathWorld, Dirichlet L-Series, accessed 20/11/2015,http://mathworld.wolfram.com/DirichletL-Series.html.[8] Ricardo Pérez Marco, Notes on the Riemann hypothesis, accessed 20/11/2015,http://garlf.maia.ub.es/Milenio/img/Riemann.pdf.[9] Michael O Rabin. Probabilistic algorithm for testing primality. Journal of numbertheory 12, 128–138, 1980.[10] Dorian Goldfeld. Gauss class number problem for imaginary quadratic fields.Bulletin of the American Mathematical Society 13, 23–37, 1985.[11] John Derbyshire. Prime obsession. The Mathematical Intelligencer 26, 55–59,2004.[12] Dorian Goldfeld. The elementary proof of the prime number theorem: An his-torical perspective. In: D. Chudnovsky, G. Chudnovsky, M. Nathanson, Numbertheory, Springer, New York, 179–192, 2003.[13] Charles Jean de La Vallée Poussin. Sur la fonction ζ(s) de Riemann et le nombredes nombres premiers inferieurs à une limite donnée, volume 59. Mémoirescouronnés de l’ Academie de Belgique 59, 1-74, 1899.[14] Arnold Walfisz. Weylsche Exponentialsummen in der neueren Zahlentheorie,Deutscher Verlag der Wissenschaften, Berlin, 1963.[15] Helge von Koch. Sur la distribution des nombres premiers. Acta Mathematica24, 159–182, 1901.[256]The Plymouth Student Scientist, 2016, 9, (2), 249-257[16] Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow. Robin’s theorem,primes, and a new elementary reformulation of the Riemann hypothesis. preprintarXiv:1110.5078, 2011.[17] Keith Briggs. Abundant numbers and the Riemann hypothesis. ExperimentalMathematics 15, 251–256, 2006.[18] Wolfram MathWorld, Robin’s Theorem, accessed 10/12/2015,http://mathworld.wolfram.com/RobinsTheorem.html.[19] Jeffrey C Lagarias. An elementary problem equivalent to the Riemann hypothe-sis. American Mathematical Monthly 109, 534–543, 2002.[20] Peter Borwein, Stephen Choi, Brendan Rooney, and Andrea Weirathmueller(Eds.). The Riemann hypothesis, Springer, New York, 2006.[21] Godfrey Harold Hardy, Sur les zéros de la fonction ζ(s) de Riemann. Comp.Rend. Acad. Sci. Paris 158, 1012–1014, 1914.[22] Brian Conrey. More than two fifths of the zeros of the Riemann zeta function areon the critical line. J. reine angew. Math 399, 1–26, 1989.[23] Hung M Bui, Brian Conrey, and Matthew P Young. More than 41% of the zerosof the zeta function are on the critical line. Acta Arith. 150, 35–64, 2011.[24] Marek Wolf. Applications of statistical mechanics in number theory. Physica A:Statistical Mechanics and its Applications 274, 149–157, 1999.[25] Andrew Odlyzko, Correspondence about the origins of the Hilbert-Polya Conjec-ture, accessed 20/11/2015, http://www.dtc.umn.edu/∼odlyzko/polya/index.html.[26] Atle Selberg. Harmonic analysis and discontinuous groups in weakly symmetricRiemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20,47–87, 1956.[27] Hugh L Montgomery. The pair correlation of zeros of the zeta function. In Proc.Symp. Pure Math. 24, pages 181–193, 1973.[28] Madan Lal Mehta. Random matrices, 3rd edition, Elsevier, Academic Press,Amsterdam, 2004.[257]

The Riemann Hypothesis

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The Riemann Hypothesis

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